Estimation of finite population mean using dual auxiliary variable for non-response using simple random sampling

1.   Introduction

In survey sampling, the appropriate use of auxiliary information is known to enhance the accuracy of an estimator of the unknown population parameter. This information (auxiliary) can be used to select a random sample using SRSWR or SRSWOR. Auxiliary information gives us a sort of technique in terms of ratio, product, regression, and other methods, it is therefore necessary to have a representative part of the population, when the population of interest is more homogeneous, than simple random sampling can be used to select units. A considerable amount of work was done on estimating the population mean by simple random sampling, a number of important references include ^{[2,5,7,11,12,13,16,17,18,19,20,22,23,24,26,27,30,32]} and the references cited therein, have suggested different types of estimators to estimate the population mean and population distribution function in the presence of non-response.

As a practical matter, one of the main problems with surveys is that they suffer from non-response, non-response has a lot of ways to happen. Examples are language problems, non-availability of response, incorrect return address and input from another person, censorship or clustering is a problem across several data. The statistician has recognized for quite few time that ignoring the stochastic nature of incompleteness or non-response may change the nature of the data. Several factors affect the non-response rate for a survey, some of these factors are the type of information collected, the official status of the investigating agency, the extent of the publicity, legal requirements of respondents, the duration of the enumerator's visit and the length of withdrawal period etc.

A great deal of work has been done on the estimation of the population mean to check non-response bias and increase efficiency of estimators by different authors. The issue of non-response in sample surveys is more common and prevalent in mail surveys than in special interview surveys. ^[10] was the first to address the issue of incomplete samples in the postal or telephone surveys. For certain related work, we refer to ^{[1,2,3,12,13,14,15,17,18,20,21,23,24,25,26,28,31,32]} and the references cited therein.

On the line of ^[11] a new family of estimators is proposed for the estimation of population mean in the presence of non-response. We will prove theoretically and numerically that the proposed family of estimators is more precise than the existing estimators.

The rest of the paper is set out as follows: In Section 2, some notations are introduced by SRS with non-responding. In Section 3, the existing estimators examined for the two non-response situations. A new family of estimators is presented in Section 4 under both non-response situations using simple random sampling. The existing and proposed estimators are theoretically compared in Section 5. In Section 6, the existing and proposed family of estimators are compared numerically. Section 7 condenses the principal discovery and culminate the document.

2.   Notations

Suppose $\Omega = \{U_{1}, U_{2}, ..., U_{N}\}$ denotes be a finite population of $N$ distinct units that is bisect into two groups, respondents and non-respondents, having sizes $N_{1}$ and $N_{2}$, where $N = N_1+N_2$. Thus we denote $\Omega_{1} = \{U_{1}, U_{2}, ..., U_{N_1}\}$ for the response group and $\Omega_{2} = \{U_{1}, U_{2}, ..., U_{N_{2}}\}$ for the non-response group. In order to estimate the population mean, a sample of $n$ is taken from the underlying population by simple random sampling without replacement (SRSWOR), and for which units $n_{1}$ are responding and $n_{2} = n-n_{1}$ are not responding. It is also assumed that the sample size $n_{1}$ is drawn from the response group of $\Omega_{1}$ and $n_{2}$ is drawn from the non-response group of $\Omega_{2}$. Moreover a sample of size $r = n_{2}/k$ units, where $k > 1$ is drawn by simple random sampling without replacement from $n_{2}$, and the temporal response is obtained from all $r$ units.

Let $Y$, $X$, $Z$, be the study, auxiliary and ranks of the auxiliary variable.

$\bar{Y} = \sum_{i = 1}^{N}Y_{i}/N$, $\hat{\bar{Y}} = \sum_{i = 1}^{n}Y_{i}/n$: The population and sample mean of $Y$.

$\bar{X} = \sum_{i = 1}^{N}X_{i}/N$, $\hat{\bar{X}} = \sum_{i = 1}^{n}X_{i}/n$: The population and sample mean of $X$.

$\bar{Z} = \sum_{i = 1}^{N}Z_{i}/N$, $\hat{\bar{Z}} = \sum_{i = 1}^{n}Z_{i}/n$: The population and sample mean of $Z$.

$\bar{Y}_{(2)} = \sum_{i = 1}^{N_2}Y_{i}/N_{2}$: The population mean of $Y$ for non-response group.

$\bar{X}_{(2)} = \sum_{i = 1}^{N_2}X_{i}/N_{2}$: The population mean of $X$ for non-response group.

$\bar{Z}_{(2)} = \sum_{i = 1}^{N_2}Z_{i}/N_{2}$: The population mean of $Z$ for non-response group.

$\hat{\bar{Y}}_{(1)} = \sum_{i = 1}^{n_{1}}Y_i/n_{1}$ denote the sample mean based on $n_{1}$ responding units out of $n$ units.

$\hat{\bar{X}}_{(1)} = \sum_{i = 1}^{n_{1}}X_i/n_{1}$ denote the sample mean based on $n_{1}$ responding units out of $n$ units.

$\hat{\bar{Z}}_{(1)} = \sum_{i = 1}^{n_{1}}Z_i/n_{1}$ be the sample mean based on $n_{1}$ responding units out of $n$ units.

$\hat{\bar{Y}}_{(2r)} = \sum_{i = 1}^{r}Y_i/r$ be the sample mean based on $r$ reacting units out of $n_{2}$ non-response units.

$\hat{\bar{X}}_{(2r)} = \sum_{i = 1}^{r}X_i/r$ be the sample mean based on $r$ reacting units out of $n_{2}$ non-response units.

$\hat{\bar{Z}}_{(2r)} = \sum_{i = 1}^{r}Z_i/r$ denote the sample mean based on $r$ reacting units out of $n_2$ non-response units.

$S^{2}_{Y} = \sum\limits_{i = 1}^N{(Y_{i}-\bar{Y})^2}/(N-1)$, $S^{2}_{X} = \sum\limits_{i = 1}^N{(X_{i}-\bar{X})^2}/(N-1)$. $S^{2}_{Z} = \sum\limits_{i = 1}^N{(Z_{i}-\bar{Z})^2}/(N-1)$: The population variance of $Y$, $X$, and $Z$.

$S^{2}_{Y_{2}} = \sum\limits_{i = 1}^{N_{2}}{(Y_{i}-\bar{Y}_{2})^2}/(N_{2}-1)$, $S^{2}_{X_{2}} = \sum\limits_{i = 1}^{N_{2}}{(X_{i}-\bar{X}_{2})^2}/(N_{2}-1)$, $S^{2}_{Z_{2}} = \sum\limits_{i = 1}^{N_{2}}{(Z_{i}-\bar{Z}_{2})^2}/(N_{2}-1)$: The population variance of $Y$, $X$, and $Z$ for non-response group.

$C_{Y} = S_{Y}/\bar{Y}$, $C_{X} = S_{X}/\bar{X}$, $C_{Z} = S_{Z}/\bar{Z}$: The population coefficient of variation of $Y$, $X$ and $Z$.

$C_{Y(2)} = S_{Y(2)}/\bar{Y}_{(2)}$, $C_{X(2)} = S_{X(2)}/\bar{X}_{(2)}$, $C_{Z(2)} = S_{Y(2)}/\bar{Z}_{(2)}$: Be the population coefficient of variation of $Y$, $X$ and $Z$ for non-response group.

$S_{YX} = \sum\limits_{i = 1}^N(Y_{i}-\bar{Y})(X_{i}-\bar{X})/(N-1)$, $S_{YZ} = \sum\limits_{i = 1}^N(Y_{i}-\bar{Y})(Z_{i}-\bar{Z})/(N-1)$, $S_{XZ} = \sum\limits_{i = 1}^N(X_{i}-\bar{X})(Z_{i}-\bar{Z})/(N-1)$: The population covariance between $(Y, X)$, $(Y, Z)$, and $(X, Z)$.

$S_{Y_{2}X_{2}} = \sum\limits_{i = 1}^{N_{2}}(Y_{i}-\bar{Y}_{2})(X_{i}-\bar{X}_{2})/(N_{2}-1)$, $S_{Y_{2}Z_{2}} = \sum\limits_{i = 1}^{N_{2}}(Y_{i}-\bar{Y}_{2})(Z_{i}-\bar{Z}_{2})/(N_{2}-1)$, $S_{X_{2}Z_{2}} = \sum\limits_{i = 1}^{N_{2}}(X_{i}-\bar{X}_{2})(Z_{i}-\bar{Z}_{2})/(N_{2}-1)$: The population covariance between $(Y, X)$, $(Y, Z)$, and $(X, Z)$ for non-response group.

$\rho_{YX} = S_{YX}/(S_{Y}S_{X})$, $\rho_{YZ} = S_{YZ}/(S_{Y}S_{Z})$, $\rho_{XZ} = S_{XZ}/(S_{X}S_{Z})$: Be the population correlation coefficient between $(Y, X)$, $(Y, Z)$, and $(X, Z)$.

$\rho_{Y_{2}X_{2}} = S_{Y_{2}X_{2}}/(S_{Y_{2}}S_{X_{2}})$, $\rho_{Y_{2}Z_{2}} = S_{Y_{2}Z_{2}}/(S_{Y_{2}}S_{Z_{2}})$, $\rho_{X_{2}Z_{2}} = S_{X_{2}Z_{2}}/(S_{X_{2}}S_{Z_{2}})$: The population correlation coefficient between $(Y, X)$, $(Y, Z)$, and $(X, Z)$ for non-response group.

$R^{2}_{Y.XZ} = (\rho^{2}_{YX}+\rho^{2}_{YZ}-2\rho_{YX}\rho_{YZ}\rho_{XZ})/(1-\rho^{2}_{XZ})$: The population coefficient of multiple determination of $Y$ on $X$ and $Z$.

$R^{2}_{Y.XZ(2)} = (\rho^{2}_{YX(2)}+\rho^{2}_{YZ(2)}-2\rho_{YX(2)}\rho_{YZ(2)}\rho_{XZ(2)})/(1-\rho^{2}_{XZ(2)})$: The population coefficient of multiple determination of $Y$ on $X$ and $Z$ for non-response group.

The population mean $Y$ may be written as such

where $W_{j} = N_{j}/N$, $\bar{Y}_j = \sum_{i = 1}^{N_{j}}Z_i/N_{j}$, for $j = 1, 2.$, $\bar{X}_j = \sum_{i = 1}^{N_{j}}X_i/N_{j}$ and $\bar{Z}_j = \sum_{i = 1}^{N_{j}}Z_i/N_{j}$. Following ^[10,12] have suggested an unbiased estimator of $\bar{Y}$ under non-response, which is given by

and

where $w_j = n_j/n$ for $j =$1, 2, $\lambda = (1/n-1/N)$ and $\lambda_2 = W_2(k-1)$.

Similarly

are unbiased estimators of $\bar{X}$ and $\bar{Z}$ respectively under non-response with corresponding variances

respectively.

In order to obtain the properties of the proposed estimator, we consider the following relative error terms.

Let $\xi^{*}_{0} = (\hat{\bar{Y}}_{H}-\bar{Y})/\bar{Y}$, $\xi^{*}_{1} = (\hat{\bar{X}}_{H}-\bar{X})/\bar{X}$, $\xi^{*}_{2} = (\hat{\bar{Z}}_{H}-\bar{Z})/\bar{Z}$, $\xi_{1} = (\hat{\bar{X}}_{H}-\bar{X})/\bar{X}$, and $\xi_{2} = (\hat{\bar{Z}}_{H}-\bar{Z})/\bar{Z}$, such that $E(\xi^{*}_{i}) = E(\xi_{i}) = 0$ for $i$ = 0, 1, 2, and for $i$ = 1, 2. Where $E(\cdot)$ represents the mathematical expectation of $(\cdot)$. Let

where $r, s, t, u = 0, 1, 2.$ Here,

where $\theta = (1/n-1/N)$ and $\theta_{2} = W_{2}(k-1)/n$.

Usually in case of non-response, two situations are more likely to happen, namely non-response on $Y$ only (say Situation-Ⅰ) and non-response on both $Y$, $X$ and $Z$ (say Situation-Ⅱ).

3.   Existing estimators

In this portion, some existing estimates of the population mean for non-response are briefly reviewed for both situations.

3.1.   Situation-Ⅰ

When non-response occurs in only one study variable, say $Y$

$(1)$ The estimator of the typical ratio of the $\bar{Y}$ is given as:

The properties of $\hat{\bar{Y}}_{R}$, are given by:

respectively.

$(2)$ The typical product estimator $\bar{Y}$ is given as:

The properties of $\hat{\bar{Y}}_{P}^*$, are given as:

$(3)$ The typical difference estimator for the $\bar{Y}$ is given as:

The minimal variance of $\hat{\bar{Y}}_{D}^*$ at $d_{\text{(opt)}} = (\bar{Y}V_{110})/(\bar{X}V_{020})$ is given as:

Here in (3.6) can be written as:

$(4)$ Following ^[27], a difference-type estimator of $\bar{Y}$ is

The properties of $\hat{\bar{Y}}_{R, D}^*$, are given by:

and

By simplify Eq (3.10) the value of $k_{1}$ and $k_{2}$, are given as:

respectively. The minimal MSE of $\hat{\bar{Y}}_{R, D}^*$ at the optimal values is given by:

Equation (3.11) may be written as

$(5)$ Following ^[4], is given as:

The biases and MSEs of $\hat{\bar{Y}}_{BT, R}^*$ and $\hat{\bar{Y}}_{BT, P}^*$, are given as:

and

$(6)$ Following ^[29], a generalized ratio-type exponential estimator of $\bar{Y}$ is

The properties of $\hat{\bar{Y}}_{S}^*$, are given as:

where $\theta = a\bar{X}/(a\bar{X}+b)$.

$(7)$ Following ^[8], a generalized class of ratio-type exponential estimators of $\bar{Y}$ is given as:

The properties of $\hat{\bar{Y}}_{GK}^*$, are given as:

The optimum values of $k_{1}$ and $k_{2}$ determined by simplifying (23), are given as:

The simplified minimum MSE of $\hat{\bar{Y}}_{GK}^*$ at the optimum values of $k_{1}$ and $k_{2}$ is given by

Here (3.20) may be written as

3.2.   Situation-Ⅱ

When non response is occur in both study and auxiliary variables, say $Y$ and $X$.

$(1)$ The traditional ratio estimator of $\bar{Y}$ is given as:

The properties of $\hat{\bar{Y}}_{R}^{**}$, are given as:

$(2)$ The traditional product estimator of $\bar{Y}$ is given as:

The properties of $\hat{\bar{Y}}_{P}^{**}$, are given as:

$(3)$ The traditional difference estimator of $\bar{Y}$ is

The minimal variance of $\hat{\bar{Y}}_{D}^{**}$ at the optimal value $d_{\text{(opt)}} = (\bar{Y}\Psi_{110})/(\bar{X}\Psi_{020})$ is

Equation (3.27) may be written as:

$(4)$ Following ^[27], a difference-type estimator of $\bar{Y}$ is

The properties of $\hat{\bar{Y}}_{R, D}^{**}$, are given as:

and

The optimal values of $k_{1}$ and $k_{2}$, determined by minimizing (3.30), are given as:

The minimal MSE of $\hat{\bar{Y}}_{R, D}^{**}$ at the optimal values is given by:

Equation (3.31) may be written as:

$(5)$ Following ^[4], the ratio and product-type exponential estimators of $\bar{Y}$, are given by:

The biases and MSEs of $\hat{\bar{Y}}_{BT, R}^{**}$ and $\hat{\bar{Y}}_{BT, P}^{**}$, are given by:

and

$(6)$ Following ^[6], a generalized ratio-type exponential estimator of $\bar{Y}$ is given by:

The properties of $\hat{\bar{Y}}_{S}^{**}$, are given by:

where $\theta = a\bar{X}/(a\bar{X}+b)$.

$(7)$ Following ^[8], estimators of $\bar{Y}$ is given by:

The properties of $\hat{\bar{Y}}_{GK}^{**}$, are given by:

The ideal values of $k_{1}$ and $k_{2}$ is expressing by (3.40),

The minimal MSE of $\hat{\bar{Y}}_{GK}$ at the optimal values of $k_{1}$ and $k_{2}$ is given by:

Equation (3.41) may be written as:

4.   Proposed estimator in non-response using simple random sampling

The proper use of ancillary variable improve the accuracy of estimator in the design and estimation stages. Complete auxiliary information is frequently supplied along with the sample frame for social, economic, and natural surveys. When the study variable and the auxiliary variable have a sufficient amount of connection, the rankings of the auxiliary variable are also correlated with the values of the auxiliary variable. Consequently, The categorised auxiliary variable (which includes the auxiliary variable's rank) can be treated as a new auxiliary variable, and this information can help an estimator perform better. Because of We present an improved family of estimators for predicting the population mean that requires additional information on the study and auxiliary variable sample means, as well as the ranks of the auxiliary variable under non-response using simple random sampling.

4.1.   Situation-Ⅰ

When non-response occur only in study variable. On the lines of ^[11], the proposed improved estimator of $\bar{Y}$ in the presence of non-response using SRS, say $\hat{\bar{Y}}_{Suggested}^*$ is given as:

where $w_{1}$, $w_{2}$, and $w_{3}$ are unknown constant. The proposed estimator $\hat{\bar{Y}}_{Suggested}^*$ can be rewritten as

Simplifying (4.2), we have

The properties of $\hat{\bar{Y}}_{Suggested}^{*}$, are given as:

The optimal values of $w_{1}$, $w_{2}$, and $w_{3}$ determined by minimizing (4.4), are

The minimal MSE of $\hat{\bar{Y}}_{Suggested}^{*}$ at optimal values of $w_{1}$, $w_{2}$ and $w_{3}$ is given by:

Equation (4.5) me be written as

where

4.2.   Situation-Ⅱ

When non-response are in both study and auxiliary variable. Taking motivation on the lines of ^[11], we proposed a family of estimators of $\bar{Y}$ in the presence of non-response say $\hat{\bar{Y}}_{Suggested}^{**}$, is given by:

where $w_{1}$, $w_{2}$, and $w_{3}$ are unknown constants. The proposed estimator $\hat{\bar{Y}}_{Suggested}^{**}$ can be rewritten as:

Simplifying (4.9), we can write

The properties of $\hat{\bar{Y}}_{Suggested}^{**}$, are given by:

The optimal values of $w_{1}$, $w_{2}$, and $w_{3}$ determined by minimizing (4.11), are given as:

The minimal MSE of the $\hat{\bar{Y}}_{Suggested}^{**}$ at optimal values of $w_{1}$, $w_{2}$ and $w_{3}$ is given by:

Here (4.12) me be written as:

where

In Table 1, we put some members of the ^[8,29], and proposed families of estimators with selected choices of $a$ and $b$.

5.   Efficiency comparisons for both condition

5.1.   Situation-Ⅰ

In this section, we performed a comparison of the adapted and proposed estimators, when non-response is available in the study variable.

(ⅰ) By taking (2.3) and (3.21),

(ⅱ) By taking (2.4) and (3.21),

(ⅲ) By taking (3.2) and (3.21),

(ⅳ) By taking (3.5) and (3.21),

(ⅴ) By taking (3.10) and (3.21),

(ⅵ) By taking (3.12) and (3.21),

(ⅶ) By taking (3.16) and (3.21),

5.2.   Situation-Ⅱ

In this section, we made efficiency comparison of all estimator, when non-response occur in both the study and auxiliary variables.

(ⅰ) By taking (2.3) and (3.21),

(ⅱ) By taking (2.4) and (3.21),

(ⅲ) By taking (3.2) and (3.21),

(ⅳ) By taking (3.5) and (3.21),

(ⅴ) By taking (3.10) and (3.21),

(ⅵ) By taking (3.12) and (3.21),

(ⅶ) By taking (3.16) and (3.21),

6.   Numerical investigation

In this section, the mathematical result is shown to verify the effectiveness of all estimators as compared to existing estimators. Four data sets are under consideration. The data description and mean square error are listed in Tables 2 and 3. The percent efficiency of estimator $\hat{\bar{Y}}_{i}$ w.r.t $\hat{\bar{Y}}_{SRS}$:

where $i = {R, P, \dots, Sugeested}$.

The MSEs and PREs of mean estimators, computed from two populations, are given in Tables 4–11.

Population Ⅰ. (Source:^[9]) $Y$: The egg assemble in 1990, $X$: Value per dozen in 1991.

Population Ⅱ. (Source:^[9]) $Y$: Eggs assemble in 1990, $X$: Value per dozen in 1990.

7.   Conclusions

In this paper, a new family of estimators for estimating the population mean with information on the auxiliary variable in the form of the sample mean and ranks of the auxiliary variable in the presence of non-response has been devised. The suggested family of estimators a mathematical expressions for biases and minimum MSEs have been generated up to the first order of approximation and compared both theoretically and numerically with the ^[6,10,22], the conventional difference, ^[8,27,29] estimators under Situation-Ⅰ and Situation-Ⅱ. It has been observed that the proposed family of estimators is more efficient in both non-response situations.

Conflict of interest

The authors declare no conflict of interest.